Dynamic Scaling in a 2+1 Dimensional Limited Mobility Model of Epitaxial Growth

We study statistical scale invariance and dynamic scaling in a simple solid-on-solid 2+1 - dimensional limited mobility discrete model of nonequilibrium surface growth, which we believe should describe the low temperature kinetic roughening properties of molecular beam epitaxy. The model exhibits long-lived ``transient'' anomalous and multiaffine dynamic scaling properties similar to that found in the corresponding 1+1 - dimensional problem. Using large-scale simulations we obtain the relevant scaling exponents, and compare with continuum theories.Comment: 5 pages, 4 ps figures included, RevTe

Records and sequences of records from random variables with a linear trend

We consider records and sequences of records drawn from discrete time series of the form $X_{n}=Y_{n}+cn$, where the $Y_{n}$ are independent and identically distributed random variables and $c$ is a constant drift. For very small and very large drift velocities, we investigate the asymptotic behavior of the probability $p_n(c)$ of a record occurring in the $n$th step and the probability $P_N(c)$ that all $N$ entries are records, i.e. that $X_1 < X_2 < ... < X_N$. Our work is motivated by the analysis of temperature time series in climatology, and by the study of mutational pathways in evolutionary biology.Comment: 21 pages, 7 figure

Non-neutral theory of biodiversity

We present a non-neutral stochastic model for the dynamics taking place in a meta-community ecosystems in presence of migration. The model provides a framework for describing the emergence of multiple ecological scenarios and behaves in two extreme limits either as the unified neutral theory of biodiversity or as the Bak-Sneppen model. Interestingly, the model shows a condensation phase transition where one species becomes the dominant one, the diversity in the ecosystems is strongly reduced and the ecosystem is non-stationary. This phase transition extend the principle of competitive exclusion to open ecosystems and might be relevant for the study of the impact of invasive species in native ecologies.Comment: 4 pages, 3 figur

Derivation of continuum stochastic equations for discrete growth models

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain--Lai--Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.Comment: 5 pages, no figur

Evolution in random fitness landscapes: the infinite sites model

We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution $g(w)$. This is the finite population version of Kingman's house of cards model [J.F.C. Kingman, \textit{J. Appl. Probab.} \textbf{15}, 1 (1978)]. In contrast to Kingman's work, the focus here is on unbounded distributions $g(w)$ which lead to an indefinite growth of the population fitness. The model is solved analytically in the limit of infinite population size $N \to \infty$ and simulated numerically for finite $N$. When the genome-wide mutation probability $U$ is small, the long time behavior of the model reduces to a point process of fixation events, which is referred to as a \textit{diluted record process} (DRP). The DRP is similar to the standard record process except that a new record candidate (a number that exceeds all previous entries in the sequence) is accepted only with a certain probability that depends on the values of the current record and the candidate. We develop a systematic analytic approximation scheme for the DRP. At finite $U$ the fitness frequency distribution of the population decomposes into a stationary part due to mutations and a traveling wave component due to selection, which is shown to imply a reduction of the mean fitness by a factor of $1-U$ compared to the $U \to 0$ limit.Comment: Dedicated to Thomas Nattermann on the occasion of his 60th birthday. Submitted to JSTAT. Error in Section 3.2 was correcte

Anisotropic Colossal Magnetoresistance Effects in Fe_{1-x}Cu_xCr_2S_4

A detailed study of the electronic transport and magnetic properties of Fe$_{1-x}$Cu$_x$Cr$_2$S$_4$ ($x \leq 0.5$) on single crystals is presented. The resistivity is investigated for $2 \leq T \leq 300$ K in magnetic fields up to 14 Tesla and under hydrostatic pressure up to 16 kbar. In addition magnetization and ferromagnetic resonance (FMR) measurements were performed. FMR and magnetization data reveal a pronounced magnetic anisotropy, which develops below the Curie temperature, $T_{\mathrm{C}}$, and increases strongly towards lower temperatures. Increasing the Cu concentration reduces this effect. At temperatures below 35 K the magnetoresistance, $MR = \frac{\rho(0) - \rho(H)}{\rho(0)}$, exhibits a strong dependence on the direction of the magnetic field, probably due to an enhanced anisotropy. Applying the field along the hard axis leads to a change of sign and a strong increase of the absolute value of the magnetoresistance. On the other hand the magnetoresistance remains positive down to lower temperatures, exhibiting a smeared out maximum with the magnetic field applied along the easy axis. The results are discussed in the ionic picture using a triple-exchange model for electron hopping as well as a half-metal utilizing a band picture.Comment: some typos correcte

Reaction Diffusion Models in One Dimension with Disorder

We study a large class of 1D reaction diffusion models with quenched disorder using a real space renormalization group method (RSRG) which yields exact results at large time. Particles (e.g. of several species) undergo diffusion with random local bias (Sinai model) and react upon meeting. We obtain the large time decay of the density of each specie, their associated universal amplitudes, and the spatial distribution of particles. We also derive the spectrum of exponents which characterize the convergence towards the asymptotic states. For reactions with several asymptotic states, we analyze the dynamical phase diagram and obtain the critical exponents at the transitions. We also study persistence properties for single particles and for patterns. We compute the decay exponents for the probability of no crossing of a given point by, respectively, the single particle trajectories ($\theta$) or the thermally averaged packets ($\bar{\theta}$). The generalized persistence exponents associated to n crossings are also obtained. Specifying to the process $A+A \to \emptyset$ or A with probabilities $(r,1-r)$, we compute exactly the exponents $\delta(r)$ and $\psi(r)$ characterizing the survival up to time t of a domain without any merging or with mergings respectively, and $\delta_A(r)$ and $\psi_A(r)$ characterizing the survival up to time t of a particle A without any coalescence or with coalescences respectively. $\bar{\theta}, \psi, \delta$ obey hypergeometric equations and are numerically surprisingly close to pure system exponents (though associated to a completely different diffusion length). Additional disorder in the reaction rates, as well as some open questions, are also discussed.Comment: 54 pages, Late